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We propose fully discrete, implicit-in-time finite-volume schemes for ageneral family of non-linear and non-local Fokker-Planck equations with agradient-flow structure, usually known as aggregation-diffusion equations, inany dimension. The schemes enjoy the positivity-preservation andenergy-dissipation properties, essential for their practical use. Thefirst-order scheme verifies these properties unconditionally for generalnon-linear diffusions and interaction potentials, while the second-order schemedoes so provided a CFL condition holds. Sweeping dimensional splitting permitsthe efficient construction of these schemes in higher dimensions whilepreserving their structural properties. Numerical experiments validate theschemes and show their ability to handle complicated phenomena typical inaggregation-diffusion equations, such as free boundaries, metastability,merging and phase transitions.

Fully Discrete Positivity-Preserving and Energy-Dissipating Schemes for Aggregation-Diffusion Equations with a Gradient Flow Structure